Lie Markov models

Sumner, Jeremy G.; Fernández-Sánchez, Jesús; Jarvis, Peter

doi:10.1016/j.jtbi.2011.12.017

Cited by 61 publications

(118 citation statements)

References 31 publications

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“…For this property to hold for a given Markov model, Sumner et al (2012a) have shown that is a sufficient condition that the subset of rate matrices that define the model be: (i) closed under addition and scalar multiplication (i.e. the set forms a vector space), and…”

Section: Composition Of the Lie Markov Modelsmentioning

confidence: 99%

“…We will give a detailed explanation of what is meant by closure and why it is of practical importance, but essentially it ensures that a nonhomogeneous process (where rate matrices change with time while staying within a given model) is equivalent to an “average” homogeneous process using rate matrices obtainable from the same model. Models which do not have this property (notably including general time-reversible model, GTR) have a consistency problem when modeling a nonhomogeneous process (Sumner et al 2012a): if a sequence evolves for a time under one set of GTR rate parameters, then for a time under a different set of GTR rate parameters, the joint probabilities (pattern frequencies) between the start and end of this process cannot (in general) be described by a single GTR model. One consequence of this is that, in a nonhomogeneous GTR model (i.e., different GTR rate matrices on each branch of a tree), pruning the tree changes the distribution of site patterns achievable at the remaining taxa.…”

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confidence: 99%

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A New Hierarchy of Phylogenetic Models Consistent with Heterogeneous Substitution Rates

2015

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When the process underlying DNA substitutions varies across evolutionary history, some standard Markov models underlying phylogenetic methods are mathematically inconsistent. The most prominent example is the general time-reversible model (GTR) together with some, but not all, of its submodels. To rectify this deficiency, nonhomogeneous Lie Markov models have been identified as the class of models that are consistent in the face of a changing process of DNA substitutions regardless of taxon sampling. Some well-known models in popular use are within this class, but are either overly simplistic (e.g., the Kimura two-parameter model) or overly complex (the general Markov model). On a diverse set of biological data sets, we test a hierarchy of Lie Markov models spanning the full range of parameter richness. Compared against the benchmark of the ever-popular GTR model, we find that as a whole the Lie Markov models perform well, with the best performing models having 8–10 parameters and the ability to recognize the distinction between purines and pyrimidines.

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Section: Composition Of the Lie Markov Modelsmentioning

confidence: 99%

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confidence: 99%

A New Hierarchy of Phylogenetic Models Consistent with Heterogeneous Substitution Rates

2015

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“…where H(t) = Q(t) T , and p(t) is a column probability vector with component p i (t) describing the probability of finding the system in state i at time t. Using the 'ket' notation |·⟩ [23,24], the probability vector can alternatively be recast as…”

Section: The Wei-norman Methodsmentioning

confidence: 99%

Lie algebraic discussion for affinity based information diffusion in social networks

Shang

2017

Open Physics

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Abstract:In this paper we develop a dynamical information diffusion model which features the affinity of people with information disseminated in social networks. Four types of agents, i.e., susceptible, informed, known, and refractory ones, are involved in the system, and the affinity mechanism composing of an affinity threshold which represents the fitness of information to be propagated is incorporated. The model can be generally described by a timeinhomogeneous Markov chain, which is governed by its master (Kolmogorov) equation. Based on the Wei-Norman method, we derive analytical solutions of the model by constructing a low-dimensional Lie algebra. Numerical examples are provided to illustrate the obtained theoretical results. This study provides useful insights into the closed-form solutions of complex social dynamics models through the Lie algebra method.

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“…The matrices involved are countably infinite in dimension, and are defined implicitly by (11) and (12). It is also possible to write explicit definitions in terms of the Kronecker delta:…”

Section: A Birth-death Processmentioning

confidence: 99%

“…The idea of combining Lie algebras and symmetry considerations with Markov chains has continued to attract theoretical interest in a variety of contexts [6], [9], [12], [15].…”

Section: Introductionmentioning

confidence: 99%

Lie Algebra Solution of Population Models Based on Time-Inhomogeneous Markov Chains

House

2012

J. Appl. Probab.

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Many natural populations are well modelled through time-inhomogeneous stochastic processes. Such processes have been analysed in the physical sciences using a method based on Lie algebras, but this methodology is not widely used for models with ecological, medical, and social applications. In this paper we present the Lie algebraic method, and apply it to three biologically well-motivated examples. The result of this is a solution form that is often highly computationally advantageous.

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Lie Markov models

Cited by 61 publications

References 31 publications

A New Hierarchy of Phylogenetic Models Consistent with Heterogeneous Substitution Rates

A New Hierarchy of Phylogenetic Models Consistent with Heterogeneous Substitution Rates

Lie algebraic discussion for affinity based information diffusion in social networks

Lie Algebra Solution of Population Models Based on Time-Inhomogeneous Markov Chains

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